Rauch comparison theorem

Rauch comparison theorem In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, qui l'a prouvé dans 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitivement, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. This theorem is formulated using Jacobi fields to measure the variation in geodesics.

Statement This section needs attention from an expert in mathematics. See the talk page for details. WikiProject Mathematics may be able to help recruit an expert. (Février 2015) Laisser {style d'affichage M,{widetilde {M}}} be Riemannian manifolds, laisser {gamma de style d'affichage :[0,J]to M} et {style d'affichage {widetilde {gamma }}:[0,J]à {widetilde {M}}} be unit speed geodesic segments such that {style d'affichage {widetilde {gamma }}(0)} has no conjugate points along {style d'affichage {widetilde {gamma }}} , et laissez {displaystyle J,{widetilde {J}}} be normal Jacobi fields along {gamma de style d'affichage } et {style d'affichage {widetilde {gamma }}} tel que {displaystyle J(0)={widetilde {J}}(0)=0} et {style d'affichage |RÉ_{t}J(0)|=gauche|{widetilde {ré}}_{t}{widetilde {J}}(0)droit|} . Suppose that the sectional curvatures of {style d'affichage M} et {style d'affichage {widetilde {M}}} satisfy {style d'affichage K(Pi )leq {widetilde {K}}({widetilde {Pi }})} chaque fois que {displaystyle Pi subset T_{gamma (t)}M} is a 2-plane containing {style d'affichage {point {gamma }}(t)} et {style d'affichage {widetilde {Pi }}subset T_{{tilde {gamma }}(t)}{widetilde {M}}} is a 2-plane containing {style d'affichage {point {widetilde {gamma }}}(t)} . Alors {style d'affichage |J(t)|gq |{widetilde {J}}(t)|} pour tous {étain de style d'affichage [0,J]} .

See also Toponogov's theorem References do Carmo, M.P. Riemannian Geometry, Birkhauser, 1992. Lee, J. M, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.

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